Does matter exist because of fields, or do fields exist due to matter? Can one exist without the other? I was wondering if fields give rise to matter or is it existence of matter which gives rise to field.
 A: All particles are actually excitations, or waves in a particular field. It is proper to think of a particle as existing due to its respective field.
The universe contains all sorts of fields like the Higgs field, the electron field, the photon field etc., and what we call "particles" are excitations of these fields. For example, an electron is simply an excitation of the electron field.
So matter, or particles, can be thought to exist as a result of fields.
A: There really isn't a right answer to this question, since fields are a mathematical construct, and it isn't ever clear when the math we use actually reflects something about reality. I will offer my perspective on the matter. Particles are the physical reality, and fields only provide a convenient way of describing them. For instance, you may know that quarks are not possible to observe on there own, but rather they can only come in quark anti-quark pairs $q\overline{q}$ or in triplets like $uud$ to make a proton for example. In the Standard Model, we use a quark 'field' to describe this unobservable particle, not the actual particles we see in experiment. So what does the quark 'field' actually describe if the quark is unobservable? The answer is it doesn't.
There is no field which describes protons (modulo effective field theory constructions). Rather, it has been deduced that the theory of QCD -is likely to- admit protons in the spectrum of particles. Of course, QCD is built out of many fields, but each of the fields do not directly correspond to the physical particles that arise in the theory.
Ultimately, you can decide whether the fields cause the particles to exist or the fields are a convenient way to describe them mathematically.
A: An experimentalist's answer:
The quantum field theory of the standard model of physics axiomatically assumes that the handful of particles and antiparticles in the table are the basic quantum level from which all other observations  (composite particles and bulk matter) could be modeled theoretically.
The success of the theory in describing experimental results leads  many theoretically inclined physicists to assume the existence of the mathematics of the model in space.
Each particle has a field all over space time, an electron field, a neutrino field ,etc , these represented as the plane wave ( i.e. no potentials) solution of the corresponding quantum mechanical equation, (Dirac for electrons for example).On these fields differential creation and annihilation operators create or annihilate  the particle.  So the existence of particles in this model depends on the fields as if they are a coordingate system, or a Lorenz invariant aether. The formulation allows to calculate interactions of more than two particles with the use of perturbation theory and Feynman diagrams.
So my answer is:
Matter exists and gives experimental data which are fitted with a theoretical model that has fields+(creation and annihilation operators) to model the interactions of elementary particles.
